mathlib documentation

algebra.group.with_one

Adjoining a zero/one to semigroups and related algebraic structures #

This file contains different results about adjoining an element to an algebraic structure which then behaves like a zero or a one. An example is adjoining a one to a semigroup to obtain a monoid. That this provides an example of an adjunction is proved in algebra.category.Mon.adjunctions.

Another result says that adjoining to a group an element zero gives a group_with_zero. For more information about these structures (which are not that standard in informal mathematics, see algebra.group_with_zero.basic)

def with_one (α : Type u_1) :
Type u_1

Add an extra element 1 to a type

Equations
def with_zero (α : Type u_1) :
Type u_1

Add an extra element 0 to a type

Equations
@[instance]
def with_one.has_one {α : Type u} :
Equations
@[instance]
def with_zero.has_zero {α : Type u} :
Equations
@[instance]
def with_zero.has_add {α : Type u} [has_add α] :
Equations
@[instance]
def with_one.has_mul {α : Type u} [has_mul α] :
Equations
@[instance]
def with_one.inhabited {α : Type u} :
Equations
@[instance]
def with_zero.inhabited {α : Type u} :
Equations
@[instance]
def with_one.nontrivial {α : Type u} [nonempty α] :
@[instance]
def with_zero.nontrivial {α : Type u} [nonempty α] :
@[instance]
def with_zero.has_coe_t {α : Type u} :
Equations
@[instance]
def with_one.has_coe_t {α : Type u} :
Equations
theorem with_one.some_eq_coe {α : Type u} {a : α} :
theorem with_zero.some_eq_coe {α : Type u} {a : α} :
@[simp]
theorem with_zero.coe_ne_zero {α : Type u} {a : α} :
a 0
@[simp]
theorem with_one.coe_ne_one {α : Type u} {a : α} :
a 1
@[simp]
theorem with_one.one_ne_coe {α : Type u} {a : α} :
1 a
@[simp]
theorem with_zero.zero_ne_coe {α : Type u} {a : α} :
0 a
theorem with_one.ne_one_iff_exists {α : Type u} {x : with_one α} :
x 1 ∃ (a : α), a = x
theorem with_zero.ne_zero_iff_exists {α : Type u} {x : with_zero α} :
x 0 ∃ (a : α), a = x
@[instance]
def with_zero.can_lift {α : Type u} :
Equations
@[instance]
def with_one.can_lift {α : Type u} :
Equations
@[simp]
theorem with_one.coe_inj {α : Type u} {a b : α} :
a = b a = b
@[simp]
theorem with_zero.coe_inj {α : Type u} {a b : α} :
a = b a = b
theorem with_one.cases_on {α : Type u} {P : with_one α → Prop} (x : with_one α) :
P 1(∀ (a : α), P a)P x
theorem with_zero.cases_on {α : Type u} {P : with_zero α → Prop} (x : with_zero α) :
P 0(∀ (a : α), P a)P x
@[instance]
def with_one.mul_one_class {α : Type u} [has_mul α] :
Equations
@[simp]
theorem with_one.coe_mul_hom_apply {α : Type u} [has_mul α] (ᾰ : α) :
def with_zero.coe_add_hom {α : Type u} [has_add α] :

coe as a bundled morphism

Equations
@[simp]
theorem with_zero.coe_add_hom_apply {α : Type u} [has_add α] (ᾰ : α) :
def with_one.coe_mul_hom {α : Type u} [has_mul α] :

coe as a bundled morphism

Equations
def with_zero.lift {α : Type u} [has_add α] {β : Type v} [add_zero_class β] :
add_hom α β (with_zero α →+ β)

Lift an add_semigroup homomorphism f to a bundled add_monoid homorphism.

Equations
def with_one.lift {α : Type u} [has_mul α] {β : Type v} [mul_one_class β] :
mul_hom α β (with_one α →* β)

Lift a semigroup homomorphism f to a bundled monoid homorphism.

Equations
@[simp]
theorem with_zero.lift_coe {α : Type u} [has_add α] {β : Type v} [add_zero_class β] (f : add_hom α β) (x : α) :
@[simp]
theorem with_one.lift_coe {α : Type u} [has_mul α] {β : Type v} [mul_one_class β] (f : mul_hom α β) (x : α) :
@[simp]
theorem with_zero.lift_zero {α : Type u} [has_add α] {β : Type v} [add_zero_class β] (f : add_hom α β) :
@[simp]
theorem with_one.lift_one {α : Type u} [has_mul α] {β : Type v} [mul_one_class β] (f : mul_hom α β) :
theorem with_zero.lift_unique {α : Type u} [has_add α] {β : Type v} [add_zero_class β] (f : with_zero α →+ β) :
theorem with_one.lift_unique {α : Type u} [has_mul α] {β : Type v} [mul_one_class β] (f : with_one α →* β) :
def with_one.map {α : Type u} {β : Type v} [has_mul α] [has_mul β] (f : mul_hom α β) :

Given a multiplicative map from α → β returns a monoid homomorphism from with_one α to with_one β

Equations
def with_zero.map {α : Type u} {β : Type v} [has_add α] [has_add β] (f : add_hom α β) :

Given an additive map from α → β returns an add_monoid homomorphism from with_zero α to with_zero β

Equations
@[simp]
@[simp]
theorem with_one.map_id {α : Type u} [has_mul α] :
@[simp]
theorem with_zero.map_comp {α : Type u} {β : Type v} [has_add α] [has_add β] {γ : Type w} [has_add γ] (f : add_hom α β) (g : add_hom β γ) :
@[simp]
theorem with_one.map_comp {α : Type u} {β : Type v} [has_mul α] [has_mul β] {γ : Type w} [has_mul γ] (f : mul_hom α β) (g : mul_hom β γ) :
@[simp]
theorem with_one.coe_mul {α : Type u} [has_mul α] (a b : α) :
a * b = (a) * b
@[simp]
theorem with_zero.coe_add {α : Type u} [has_add α] (a b : α) :
(a + b) = a + b
@[instance]
def with_zero.has_one {α : Type u} [one : has_one α] :
Equations
@[simp]
theorem with_zero.coe_one {α : Type u} [has_one α] :
1 = 1
@[instance]
def with_zero.mul_zero_class {α : Type u} [has_mul α] :
Equations
@[simp]
theorem with_zero.coe_mul {α : Type u} [has_mul α] {a b : α} :
a * b = (a) * b
@[simp]
theorem with_zero.zero_mul {α : Type u} [has_mul α] (a : with_zero α) :
0 * a = 0
@[simp]
theorem with_zero.mul_zero {α : Type u} [has_mul α] (a : with_zero α) :
a * 0 = 0
def with_zero.inv {α : Type u} [has_inv α] (x : with_zero α) :

Given an inverse operation on α there is an inverse operation on with_zero α sending 0 to 0

Equations
@[instance]
def with_zero.has_inv {α : Type u} [has_inv α] :
Equations
@[simp]
theorem with_zero.coe_inv {α : Type u} [has_inv α] (a : α) :
@[simp]
theorem with_zero.inv_zero {α : Type u} [has_inv α] :
0⁻¹ = 0
@[simp]
theorem with_zero.inv_one {α : Type u} [group α] :
1⁻¹ = 1
@[instance]
def with_zero.group_with_zero {α : Type u} [group α] :

if G is a group then with_zero G is a group with zero.

Equations
theorem with_zero.div_coe {α : Type u} [group α] (a b : α) :